\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  -47.0 & -4.00 x_{1} & +  6.00 x_{2} & +  7.00 x_{3} & + 10.00 x_{4} & +  7.00 x_{5}\\
 x_{7}   &  55.0 & -2.00 x_{1} & -7.00 x_{2} & -7.00 x_{3} & -8.00 x_{4} & +  2.00 x_{5}\\
 x_{8}   &  13.0 & +  8.00 x_{1} & -5.00 x_{2} & -1.00 x_{3} & -8.00 x_{4} & +  7.00 x_{5}\\
 x_{9}   &  20.0 & +  3.00 x_{1} &   & -4.00 x_{3} & -5.00 x_{4} & -3.00 x_{5}\\
 x_{10}   &  -11.0 & +  4.00 x_{1} & +  4.00 x_{2} & +  3.00 x_{3} & +  5.00 x_{4} & -10.00 x_{5}\\
 x_{11}   &  -27.0 & +  7.00 x_{1} & -6.00 x_{2} & + 10.00 x_{3} & +  2.00 x_{4} & +  7.00 x_{5}\\
 x_{12}   &  16.0 & -10.00 x_{1} & -5.00 x_{2} & -2.00 x_{3} & +  8.00 x_{4} & -3.00 x_{5}\\
 x_{13}   &  -8.0 & +  4.00 x_{1} & +  4.00 x_{2} & -9.00 x_{3} & +  6.00 x_{4} & +  5.00 x_{5}\\
\hline
z    &  0.0 & -1.00 x_{1} & +  4.00 x_{2} & -3.00 x_{3} & +  5.00 x_{4} & -5.00 x_{5}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  4.00 y_{6} & +  2.00 y_{7} & -8.00 y_{8} & -3.00 y_{9} & -4.00 y_{10} & -7.00 y_{11} & + 10.00 y_{12} & -4.00 y_{13}\\
 y_{2}   &  1.0 & -6.00 y_{6} & +  7.00 y_{7} & +  5.00 y_{8} &   & -4.00 y_{10} & +  6.00 y_{11} & +  5.00 y_{12} & -4.00 y_{13}\\
 y_{3}   &  1.0 & -7.00 y_{6} & +  7.00 y_{7} & +  1.00 y_{8} & +  4.00 y_{9} & -3.00 y_{10} & -10.00 y_{11} & +  2.00 y_{12} & +  9.00 y_{13}\\
 y_{4}   &  1.0 & -10.00 y_{6} & +  8.00 y_{7} & +  8.00 y_{8} & +  5.00 y_{9} & -5.00 y_{10} & -2.00 y_{11} & -8.00 y_{12} & -6.00 y_{13}\\
 y_{5}   &  1.0 & -7.00 y_{6} & -2.00 y_{7} & -7.00 y_{8} & +  3.00 y_{9} & + 10.00 y_{10} & -7.00 y_{11} & +  3.00 y_{12} & -5.00 y_{13}\\
\hline
z    &  -0 & + 47.00 y_{6} & -55.00 y_{7} & -13.00 y_{8} & -20.00 y_{9} & + 11.00 y_{10} & + 27.00 y_{11} & -16.00 y_{12} & +  8.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 7 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{13}   &  0.0236035232467 & -0.06 y_{4} & -0.08 y_{7} & -0.02 y_{5} & +  0.02 y_{2} & +  0.06 y_{3} & -0.08 y_{8} & -0.33 y_{12} & -0.03 y_{1}\\
 y_{9}   &  0.34280298807 & +  0.23 y_{4} & +  0.84 y_{7} & -0.08 y_{5} & -0.29 y_{2} & -0.08 y_{3} & -1.87 y_{8} & +  4.96 y_{12} & -0.13 y_{1}\\
 y_{10}   &  0.0303712788494 & -0.05 y_{4} & +  0.64 y_{7} & +  0.06 y_{5} & -0.00 y_{2} & -0.01 y_{3} & +  0.55 y_{8} & -0.20 y_{12} & -0.03 y_{1}\\
 y_{6}   &  0.223492028097 & +  0.07 y_{4} & +  0.92 y_{7} & -0.05 y_{5} & -0.16 y_{2} & -0.06 y_{3} & -0.22 y_{8} & +  1.84 y_{12} & -0.02 y_{1}\\
 y_{11}   &  0.0928085628275 & +  0.00 y_{4} & +  0.13 y_{7} & -0.02 y_{5} & +  0.02 y_{2} & -0.03 y_{3} & -0.74 y_{8} & +  0.66 y_{12} & -0.06 y_{1}\\
\hline
z    &  6.67680900881 & -2.10 y_{4} & -18.65 y_{7} & -1.14 y_{5} & -1.14 y_{2} & -1.85 y_{3} & -0.19 y_{8} & -15.56 y_{12} & -0.45 y_{1}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.1036458914 & +  0.06 x_{13} & -0.23 x_{9} & +  0.05 x_{10} & -0.07 x_{6} & -0.00 x_{11}\\
 x_{7}   &  18.6519455904 & +  0.08 x_{13} & -0.84 x_{9} & -0.64 x_{10} & -0.92 x_{6} & -0.13 x_{11}\\
 x_{5}   &  1.14003790835 & +  0.02 x_{13} & +  0.08 x_{9} & -0.06 x_{10} & +  0.05 x_{6} & +  0.02 x_{11}\\
 x_{2}   &  1.13619132568 & -0.02 x_{13} & +  0.29 x_{9} & +  0.00 x_{10} & +  0.16 x_{6} & -0.02 x_{11}\\
 x_{3}   &  1.85035120972 & -0.06 x_{13} & +  0.08 x_{9} & +  0.01 x_{10} & +  0.06 x_{6} & +  0.03 x_{11}\\
 x_{8}   &  0.192451778348 & +  0.08 x_{13} & +  1.87 x_{9} & -0.55 x_{10} & +  0.22 x_{6} & +  0.74 x_{11}\\
 x_{12}   &  15.5615676218 & +  0.33 x_{13} & -4.96 x_{9} & +  0.20 x_{10} & -1.84 x_{6} & -0.66 x_{11}\\
 x_{1}   &  0.446582673654 & +  0.03 x_{13} & +  0.13 x_{9} & +  0.03 x_{10} & +  0.02 x_{6} & +  0.06 x_{11}\\
\hline
z    &  -6.67680900881 & -0.02 x_{13} & -0.34 x_{9} & -0.03 x_{10} & -0.22 x_{6} & -0.09 x_{11}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.1036458914 & +  0.06 x_{13} & -0.23 x_{9} & +  0.05 x_{10} & -0.07 x_{6} & -0.00 x_{11}\\
 x_{7}   &  18.6519455904 & +  0.08 x_{13} & -0.84 x_{9} & -0.64 x_{10} & -0.92 x_{6} & -0.13 x_{11}\\
 x_{5}   &  1.14003790835 & +  0.02 x_{13} & +  0.08 x_{9} & -0.06 x_{10} & +  0.05 x_{6} & +  0.02 x_{11}\\
 x_{2}   &  1.13619132568 & -0.02 x_{13} & +  0.29 x_{9} & +  0.00 x_{10} & +  0.16 x_{6} & -0.02 x_{11}\\
 x_{3}   &  1.85035120972 & -0.06 x_{13} & +  0.08 x_{9} & +  0.01 x_{10} & +  0.06 x_{6} & +  0.03 x_{11}\\
 x_{8}   &  0.192451778348 & +  0.08 x_{13} & +  1.87 x_{9} & -0.55 x_{10} & +  0.22 x_{6} & +  0.74 x_{11}\\
 x_{12}   &  15.5615676218 & +  0.33 x_{13} & -4.96 x_{9} & +  0.20 x_{10} & -1.84 x_{6} & -0.66 x_{11}\\
 x_{1}   &  0.446582673654 & +  0.03 x_{13} & +  0.13 x_{9} & +  0.03 x_{10} & +  0.02 x_{6} & +  0.06 x_{11}\\
\hline
z    &  3.36516891515 & +  0.27 x_{13} & -0.72 x_{9} & +  0.47 x_{10} & -0.18 x_{6} & -0.36 x_{11}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.1036458914 & +  0.06 x_{13} & -0.23 x_{9} & +  0.05 x_{10} & -0.07 x_{6} & -0.00 x_{11}\\
 x_{7}   &  18.6519455904 & +  0.08 x_{13} & -0.84 x_{9} & -0.64 x_{10} & -0.92 x_{6} & -0.13 x_{11}\\
 x_{5}   &  1.14003790835 & +  0.02 x_{13} & +  0.08 x_{9} & -0.06 x_{10} & +  0.05 x_{6} & +  0.02 x_{11}\\
 x_{2}   &  1.13619132568 & -0.02 x_{13} & +  0.29 x_{9} & +  0.00 x_{10} & +  0.16 x_{6} & -0.02 x_{11}\\
 x_{3}   &  1.85035120972 & -0.06 x_{13} & +  0.08 x_{9} & +  0.01 x_{10} & +  0.06 x_{6} & +  0.03 x_{11}\\
 x_{8}   &  0.192451778348 & +  0.08 x_{13} & +  1.87 x_{9} & -0.55 x_{10} & +  0.22 x_{6} & +  0.74 x_{11}\\
 x_{12}   &  15.5615676218 & +  0.33 x_{13} & -4.96 x_{9} & +  0.20 x_{10} & -1.84 x_{6} & -0.66 x_{11}\\
 x_{1}   &  0.446582673654 & +  0.03 x_{13} & +  0.13 x_{9} & +  0.03 x_{10} & +  0.02 x_{6} & +  0.06 x_{11}\\
\hline
z    &  3.36516891515 & +  0.27 x_{13} & -0.72 x_{9} & +  0.47 x_{10} & -0.18 x_{6} & -0.36 x_{11}\\
\end{array}\]


 $ x_{10} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.1195525331 & +  0.07 x_{13} & -0.07 x_{9} & -0.08 x_{8} & -0.05 x_{6} & +  0.06 x_{11}\\
 x_{7}   &  18.4286950223 & -0.02 x_{13} & -3.01 x_{9} & +  1.16 x_{8} & -1.17 x_{6} & -0.98 x_{11}\\
 x_{5}   &  1.11908977506 & +  0.01 x_{13} & -0.13 x_{9} & +  0.11 x_{8} & +  0.03 x_{6} & -0.06 x_{11}\\
 x_{2}   &  1.13669470041 & -0.02 x_{13} & +  0.30 x_{9} & -0.00 x_{8} & +  0.16 x_{6} & -0.02 x_{11}\\
 x_{3}   &  1.85443241721 & -0.06 x_{13} & +  0.12 x_{9} & -0.02 x_{8} & +  0.07 x_{6} & +  0.05 x_{11}\\
 x_{10}   &  0.347289847491 & +  0.15 x_{13} & +  3.38 x_{9} & -1.80 x_{8} & +  0.39 x_{6} & +  1.33 x_{11}\\
 x_{12}   &  15.6309404048 & +  0.36 x_{13} & -4.28 x_{9} & -0.36 x_{8} & -1.76 x_{6} & -0.39 x_{11}\\
 x_{1}   &  0.45758721983 & +  0.04 x_{13} & +  0.24 x_{9} & -0.06 x_{8} & +  0.03 x_{6} & +  0.10 x_{11}\\
\hline
z    &  3.5282081204 & +  0.34 x_{13} & +  0.86 x_{9} & -0.85 x_{8} & -0.00 x_{6} & +  0.26 x_{11}\\
\end{array}\]


 $ x_{9} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  1.84853397237 & +  0.06 x_{13} & +  0.02 x_{12} & -0.08 x_{8} & -0.02 x_{6} & +  0.07 x_{11}\\
 x_{7}   &  7.43870876797 & -0.27 x_{13} & +  0.70 x_{12} & +  1.41 x_{8} & +  0.07 x_{6} & -0.71 x_{11}\\
 x_{5}   &  0.653143501551 & -0.00 x_{13} & +  0.03 x_{12} & +  0.12 x_{8} & +  0.08 x_{6} & -0.04 x_{11}\\
 x_{2}   &  2.21401184099 & +  0.00 x_{13} & -0.07 x_{12} & -0.03 x_{8} & +  0.04 x_{6} & -0.04 x_{11}\\
 x_{3}   &  2.27719199323 & -0.05 x_{13} & -0.03 x_{12} & -0.03 x_{8} & +  0.02 x_{6} & +  0.03 x_{11}\\
 x_{10}   &  12.6806174232 & +  0.43 x_{13} & -0.79 x_{12} & -2.09 x_{8} & -1.00 x_{6} & +  1.02 x_{11}\\
 x_{9}   &  3.65045108542 & +  0.08 x_{13} & -0.23 x_{12} & -0.08 x_{8} & -0.41 x_{6} & -0.09 x_{11}\\
 x_{1}   &  1.32043980829 & +  0.06 x_{13} & -0.06 x_{12} & -0.08 x_{8} & -0.07 x_{6} & +  0.08 x_{11}\\
\hline
z    &  6.68098393008 & +  0.41 x_{13} & -0.20 x_{12} & -0.92 x_{8} & -0.36 x_{6} & +  0.18 x_{11}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{7} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.53524804178 & +  0.03 x_{13} & +  0.08 x_{12} & +  0.05 x_{8} & -0.02 x_{6} & -0.09 x_{7}\\
 x_{11}   &  10.5444796718 & -0.38 x_{13} & +  1.00 x_{12} & +  2.00 x_{8} & +  0.10 x_{6} & -1.42 x_{7}\\
 x_{5}   &  0.18780305856 & +  0.01 x_{13} & -0.01 x_{12} & +  0.03 x_{8} & +  0.08 x_{6} & +  0.06 x_{7}\\
 x_{2}   &  1.74748228273 & +  0.02 x_{13} & -0.11 x_{12} & -0.12 x_{8} & +  0.04 x_{6} & +  0.06 x_{7}\\
 x_{3}   &  2.64621409922 & -0.06 x_{13} & +  0.01 x_{12} & +  0.04 x_{8} & +  0.02 x_{6} & -0.05 x_{7}\\
 x_{10}   &  23.4022752704 & +  0.04 x_{13} & +  0.22 x_{12} & -0.05 x_{8} & -0.90 x_{6} & -1.44 x_{7}\\
 x_{9}   &  2.68211488251 & +  0.12 x_{13} & -0.33 x_{12} & -0.27 x_{8} & -0.42 x_{6} & +  0.13 x_{7}\\
 x_{1}   &  2.16887355464 & +  0.03 x_{13} & +  0.02 x_{12} & +  0.08 x_{8} & -0.06 x_{6} & -0.11 x_{7}\\
\hline
z    &  8.6196381947 & +  0.34 x_{13} & -0.02 x_{12} & -0.55 x_{8} & -0.34 x_{6} & -0.26 x_{7}\\
\end{array}\]


 $ x_{13} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  3.49563341937 & -0.09 x_{11} & +  0.17 x_{12} & +  0.24 x_{8} & -0.01 x_{6} & -0.22 x_{7}\\
 x_{13}   &  27.9868825796 & -2.65 x_{11} & +  2.65 x_{12} & +  5.32 x_{8} & +  0.26 x_{6} & -3.76 x_{7}\\
 x_{5}   &  0.534526040378 & -0.03 x_{11} & +  0.02 x_{12} & +  0.10 x_{8} & +  0.08 x_{6} & +  0.02 x_{7}\\
 x_{2}   &  2.25004419616 & -0.05 x_{11} & -0.07 x_{12} & -0.02 x_{8} & +  0.04 x_{6} & -0.00 x_{7}\\
 x_{3}   &  0.929003288194 & +  0.16 x_{11} & -0.15 x_{12} & -0.29 x_{8} & +  0.01 x_{6} & +  0.18 x_{7}\\
 x_{10}   &  24.6213980129 & -0.12 x_{11} & +  0.34 x_{12} & +  0.18 x_{8} & -0.89 x_{6} & -1.61 x_{7}\\
 x_{9}   &  5.97822013223 & -0.31 x_{11} & -0.01 x_{12} & +  0.36 x_{8} & -0.39 x_{6} & -0.31 x_{7}\\
 x_{1}   &  2.92532616766 & -0.07 x_{11} & +  0.10 x_{12} & +  0.23 x_{8} & -0.05 x_{6} & -0.22 x_{7}\\
\hline
z    &  18.0933776474 & -0.90 x_{11} & +  0.88 x_{12} & +  1.25 x_{8} & -0.25 x_{6} & -1.53 x_{7}\\
\end{array}\]


 $ x_{8} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  4.26105159463 & +  0.04 x_{11} & +  0.05 x_{12} & -0.82 x_{3} & -0.00 x_{6} & -0.07 x_{7}\\
 x_{13}   &  45.1928333949 & +  0.36 x_{11} & -0.22 x_{12} & -18.52 x_{3} & +  0.38 x_{6} & -0.41 x_{7}\\
 x_{5}   &  0.848540820096 & +  0.02 x_{11} & -0.03 x_{12} & -0.34 x_{3} & +  0.08 x_{6} & +  0.08 x_{7}\\
 x_{2}   &  2.18335180397 & -0.06 x_{11} & -0.05 x_{12} & +  0.07 x_{3} & +  0.04 x_{6} & -0.02 x_{7}\\
 x_{8}   &  3.23543898535 & +  0.57 x_{11} & -0.54 x_{12} & -3.48 x_{3} & +  0.02 x_{6} & +  0.63 x_{7}\\
 x_{10}   &  25.2036694988 & -0.01 x_{11} & +  0.24 x_{12} & -0.63 x_{3} & -0.88 x_{6} & -1.49 x_{7}\\
 x_{9}   &  7.13692894964 & -0.11 x_{11} & -0.21 x_{12} & -1.25 x_{3} & -0.38 x_{6} & -0.09 x_{7}\\
 x_{1}   &  3.66260312769 & +  0.06 x_{11} & -0.03 x_{12} & -0.79 x_{3} & -0.05 x_{6} & -0.07 x_{7}\\
\hline
z    &  22.1333579608 & -0.19 x_{11} & +  0.21 x_{12} & -4.35 x_{3} & -0.22 x_{6} & -0.75 x_{7}\\
\end{array}\]


 $ x_{12} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  4.5361638819 & +  0.09 x_{11} & -0.09 x_{8} & -1.12 x_{3} & -0.00 x_{6} & -0.02 x_{7}\\
 x_{13}   &  43.8953994049 & +  0.13 x_{11} & +  0.40 x_{8} & -17.12 x_{3} & +  0.37 x_{6} & -0.66 x_{7}\\
 x_{5}   &  0.646372167544 & -0.01 x_{11} & +  0.06 x_{8} & -0.12 x_{3} & +  0.08 x_{6} & +  0.04 x_{7}\\
 x_{2}   &  1.8560311284 & -0.12 x_{11} & +  0.10 x_{8} & +  0.42 x_{3} & +  0.04 x_{6} & -0.08 x_{7}\\
 x_{12}   &  6.01396200504 & +  1.05 x_{11} & -1.86 x_{8} & -6.47 x_{3} & +  0.04 x_{6} & +  1.17 x_{7}\\
 x_{10}   &  26.6636530098 & +  0.24 x_{11} & -0.45 x_{8} & -2.20 x_{3} & -0.87 x_{6} & -1.21 x_{7}\\
 x_{9}   &  5.89688715953 & -0.33 x_{11} & +  0.38 x_{8} & +  0.09 x_{3} & -0.39 x_{6} & -0.33 x_{7}\\
 x_{1}   &  3.50560769055 & +  0.03 x_{11} & +  0.05 x_{8} & -0.62 x_{3} & -0.05 x_{6} & -0.10 x_{7}\\
\hline
z    &  23.3674753948 & +  0.03 x_{11} & -0.38 x_{8} & -5.68 x_{3} & -0.21 x_{6} & -0.51 x_{7}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  5.98823529412 & -0.78 x_{2} & -0.01 x_{8} & -0.79 x_{3} & +  0.03 x_{6} & -0.08 x_{7}\\
 x_{13}   &  46.0352941176 & -1.15 x_{2} & +  0.52 x_{8} & -16.64 x_{3} & +  0.41 x_{6} & -0.75 x_{7}\\
 x_{5}   &  0.435294117647 & +  0.11 x_{2} & +  0.05 x_{8} & -0.17 x_{3} & +  0.08 x_{6} & +  0.05 x_{7}\\
 x_{11}   &  15.9 & -8.57 x_{2} & +  0.87 x_{8} & +  3.63 x_{3} & +  0.33 x_{6} & -0.70 x_{7}\\
 x_{12}   &  22.7764705882 & -9.03 x_{2} & -0.95 x_{8} & -2.64 x_{3} & +  0.39 x_{6} & +  0.44 x_{7}\\
 x_{10}   &  30.5176470588 & -2.08 x_{2} & -0.24 x_{8} & -1.32 x_{3} & -0.79 x_{6} & -1.38 x_{7}\\
 x_{9}   &  0.7 & +  2.80 x_{2} & +  0.10 x_{8} & -1.10 x_{3} & -0.50 x_{6} & -0.10 x_{7}\\
 x_{1}   &  3.98235294118 & -0.26 x_{2} & +  0.07 x_{8} & -0.52 x_{3} & -0.04 x_{6} & -0.12 x_{7}\\
\hline
z    &  23.7823529412 & -0.22 x_{2} & -0.36 x_{8} & -5.58 x_{3} & -0.21 x_{6} & -0.52 x_{7}\\
\end{array}\]
Final Dictionary
Solution:  23.7823529412
Num Pivots:  8
\end{document}
